Development of a Negative Stiffness Bistable Damper for Structural Vibration Control

Linear dampers have been widely applied for suppressing the dynamic responses of structures to mitigate their damage. However, the primary disadvantage of the classical linear damper is that it is vulnerable to detuning, which has become an issue of great importance recently due to a great reduction in vibration control performance. To overcome the shortcoming, this study develops a negative stifness bistable damper (NSBD) composed of a simple assembly consisting of a bistable buckling beam with a mass. Energy is dissipated through the transformation between the bistable states. Te constitutive equation of the NSBD is derived to analyze the efects of the stifness ratio, the arch-span ratio, and the damping ratio on its restoring capabilities. Te vibration reduction efect of the NSBD is experimentally evaluated under diferent sinusoidal and seismic excitations in shaking table tests. Te obtained results reveal that the NSBD can efectively restrain structural displacements.


Introduction
Vibration control is well-established since frst proposed by Yao [1] in 1972, yet still developed to enhance the functionality and safety of structures. Up to now, many structures have been installed with various types of vibration control devices to restrain their responses to wind loads, seismic action, or other dynamic excitations [2][3][4][5]. Linear dampers (e.g., tuned mass damper) have been widely applied in engineering practice, such as the vibration control systems of the Akashi Kaikyo Bridge [6], the Taipei 101 tower [7], and the Shanghai Center Tower [8].
Generally, the dynamic response of a structure is greatly infuenced by the frst frequency and vibration mode [9][10][11]. Te conventional tuned mass damper (TMD) must keep consistent with the frst frequency of the primary structure to efectively restrain the structural vibration. Terefore, as a type of frequency-sensitive device, the conventional TMD operates within a relatively narrow frequency band for vibration reduction. Te TMD is detuned and its efectiveness can be evidently reduced when the natural frequency of the host structure alters.
To overcome this defciency, many negative stifness vibration control devices have been developed. Te concept of negative stifness was frst introduced by Molyneaux [12]. When the directions of restoring force and corresponding deformation are opposite, their ratio is negative. A structure in such a state has negative stifness. For structural vibration control, the negative stifness can reduce the stifness of the control system and thus lower its natural frequency, allowing it to provide vibration reduction over a wide frequency range. Negative stifness can be achieved via various methods. Traditionally, negative stifness is achieved by special mechanical devices for energy storage elements (e.g., spring and prebuckled beam) [13][14][15][16][17], magnetic elements [18][19][20], geometric nonlinearities [21][22][23], or composite structures and metamaterials [23,24]. Virgin et al. [25] proposed negative vibration isolation using a highly deformed, slender beam. Te isolator had a high adjustable stifness and a high rate of displacement transmission over a wide range. Shi and Zhu [26] developed and experimentally verifed two confgurations of magnetic negative stifness dampers (MNSDs), in which multiple permanent magnets are placed in a conductive tube. Zhou et al. [27]. introduced a negative stifness vibration isolation damper (NSVID) using a nonlinear structure and utilized the NSVID to control torsional vibration along a shaft. Tis proposed device can realize quasi-zero stifness (QZS) by changing the stifness of rubber and control parameters. Xiang et al. [28] designed a semiactive control torsional vibration damper, which is composed of positive and negative stifness in parallel. Tey provided a derivation of the theory behind the damper mechanism and analyzed its elastic properties. Zhou et al. [29] conducted theoretical analysis and experimental studies on a passive negative stifness damper (PNSD) in series with fexible support. Te results indicated that their proposed device can achieve a range of equivalent negative stifnesses and Coulomb values. Haghpanah et al. [24] invented a negative stifness element composed of one convex elastic element together with elastic elements that have nonconvex strain energy. Tis element can greatly enhance damping performance under dynamic excitation. Recently, numerous other negative stifness dampers have been introduced for vibration control of various types of excitations [30][31][32][33][34] and diferent types of structures [35][36][37].
Bistable structures are structures that not only have negative stifness properties, but can also provide cubic stifness to enhance vibration control. Most previous studies focused on mechanical properties analysis [38,39] and vibration isolation applications of bistable structures. Johnson et al. [40] studied the vibration-attenuating efect of a bistable oscillator and investigated its snap-through dynamics using the harmonic balance method and experimentation. Experimental results show that the bistable oscillator has a slightly superior vibration attenuation performance compared to the fundamental harmonic snapthrough action. Farhangdoust et al. [41]. introduced a bistable tuned mass damper (BTMD) mechanism for suppressing the vortex-induced vibration (VIV) in suspension bridge decks, which are exceptionally sensitive to broadband input of vortex shedding velocity. Xia et al. [42] investigated the enhancement in vibration attenuation bandwidth of a locally resonant metamaterial beam with bistable attachments, the interwell oscillations of the attachments produced an attenuation frequency range of 350% wider than linear locally resonant bandgap for moderate-to-high amplitude excitation levels. Zhao et al. [43] explored multiple snapthrough pathways and force-displacement curves of a discontinuous point and verifed the results through theoretical analysis and numerical simulation. Liu et al. [44] used equivalent, analytical, numerical, and experimental methods to investigate 1/2 sub-harmonic resonance in a bistable structure and its vibration isolation characteristics. Zhang et al. [45] designed a recoverable inclined beam energy absorption structure that leverages bistable characteristics to improve the safety and efciency of passive energy absorption systems in automobiles. Yan et al. [46,47] invented a bistable vibration isolation (BVI) device composed of several ring permanent magnets (PMS). Te PMS improved the vibration isolation performance of the BVI through nonlinear electromagnetic shunt damping. Zhang et al. [48] utilized an optimized and varying sectional profle to improve the performance of a quasi-zero stifness (QZS) isolation system using the nonlinear and negative stifness generated by the snap-through efect of bistable structures. Yang et al. [49] proposed a feedback control law to modulate the current input to the actuator of an active vibration isolation system. Te proposed control law greatly attenuated the transmissibility of the bistable nonlinear electromagnetic actuator from the base excitation.
Inspired by the advantages of negative stifness and bistable structures, this paper proposes a negative stifness bistable damper (NSBD) composed of a bistable buckling beam assembled with a mass. Te NSBD can enhance energy dissipation through the transformation between the bistable states. Te key component of the damper is a precompressed spring steel set in a clamped-clamped confguration in the frame structure. Te constitutive equation of the NSBD is derived to analyze the efects of the stifness ratio, the arch-span ratio, and the damping ratio on the restoring force of the NSBD. To validate the vibration reduction efect of the NSBD, a frame structure was subjected to shake table tests with and without the NSBD. Finally, a following discussion of the experimental results is presented to explore the damping efect of diferent parameters on the NSBD.

Mechanical Characterization of Negative Stifness Bistable
Damper. A schematic of a linear single-degree-of-freedom (SDOF) structure equipped with an NSBD is depicted in Figure 1. Te mass, stifness, and damping coefcient of the structure are m 1 , k 1 , and c 1 , respectively. In the schematic, k ns is the stifness of the NSBD and consists of the negative stifness k 21 and the cubic stifness k 22 . Te key component of the NSBD is the bistable buckling beam, which is clamped within the frame structure and precompressed. An attached mass is fxed at the midspan of the bistable buckling beam.
When an external excitation acts on the primary structure, the NSBD will be continuously transformed between two equilibrium positions. Under small transverse displacement, the prestress imposed along the axial direction (y direction) of the straight beam to move the right end of the beam from position b to position a results in displacement w y in the y direction. Te prestress also causes the force P to be applied in the x-direction at the midpoint of the beam, resulting in displacement w x in the x-direction, as shown in Figure 2. Te defection curve of the deformed beam is the frst-order mode shape of the clamped-clamped beam.
Assuming that the beam section thickness is δ and the beam section width is b, the moment of inertia and crosssectional area of the beam section are, respectively, I � (bδ 3 /12) and A � bδ. Te axial stifness of the beam is k 0 � (AE/l c ). According to Jin et al. [50], the relationship between the external force on the buckling beam and the displacement is as follows: 2 Shock and Vibration where F 0 � (pl 3 c /EIδ), Δ � (σ/δ), and Q � (w y /δ) are normalized parameters. l c is the span of the buckling beam and δ is the thickness. σ is the intermediate deformation of the buckling beam under external force, and w y is the arching height of the buckling beam.
Te model parameters of the bistable buckling beam are solved by equation (2) and transformed into the ordinary form: Te formula (5) can be obtained by placing the zero position at the origin.

Stifness Analysis.
From the above analysis, it can be seen that the stifness expression of the buckling beam is as follows: Te axial stifness of the buckling beam is Ten, the equivalent stifness of the buckling beam is Te variation of equivalent stifness of buckling beam with diferent camber heights is shown in Figure 4.

Mechanical Constitutive Equation of Buckling Beam.
From the above derivation, it can be seen that the mechanical constitutive equation of the buckling beam can be composed of a negative stifness term and a cubic nonlinear term as follows: where k 21 and k 22 are, respectively, the negative stifness coefcient and cubic stifness coefcient. Terefore, the potential energy function of the buckling beam is as follows: Letting F � 0, the equilibrium position of the buckling beam can be obtained as shown in equation (11).
Although the stifness of the buckling beam changes with a change in position during vibration, the response characteristics are approximately linear when the buckling beam vibrates at the equilibrium point with a very small amplitude. Terefore, the stifness of the buckling beam is simplifed with the linear stifness at the equilibrium position in the parametric analysis. In the analysis, the equivalent stifness is used for simplifcation. Te equivalent stifness is as follows: Te equivalent frequency of the buckled beam is From the potential energy function of the buckling beam, the bistable limit position can be solved as  It can be seen that when the applied external load meets the condition A 0 > x cr , the buckling beam is in bistable vibration, and when A 0 ≤ x cr , the buckling beam experiencing monostable vibration.

Analytical Derivation.
When the buckling beam with fxed ends is subjected to sinusoidal excitation with amplitude F and frequency ω, the beam can be simplifed into an SDOF system. Te dynamic equation of the buckled beam is as follows: where m, c,and k 21 , k 22 refer to the mass, damping, and negative stifness and cubic stifness of the buckling beam, respectively; the equation (16) can be simplifed to In order to obtain an approximate solution, the harmonic balance method is adopted in the paper, as used to solve equation (16); the frst-order approximate solution is assumed to be as follows: In the analysis, the frst-order approximate solution is considered, by substituting equation (17) into equation (16), and approximating sin 3 θ � (3/4)sin θ − (1/4)sin 3θ ≈ (3/4)sin θ, the following relationships are obtained: 2ξθY � − sin φ.
By combined equations (18a) and (18b) and eliminating φ, equation (19) can be obtained as follows: 9 16 Rearranging equation (19) to be expressed in terms of θ is Te solution of the equation is thus Te amplitude-frequency response function of the buckled beam can be obtained by solving equation (21), as shown in Figure 5.
Te amplitude-frequency response curve of a linear system is constant; however, the amplitude-frequency response curve of the nonlinear system depends on the initial value conditions. Figure 5 reveals the relation between the amplitude of the response of the nonlinear system and the excitation frequency at the following initial conditions: (a) ξ � 0.03, α � 0.3, (b) ξ � 0.03, f � 0.5, and (c) f � 0.5, α � 0.3. It can be noted that the negative stifness term and the nonlinear term make the curve bend towards the right, resulting in one frequency value corresponding to multiple amplitudes in the frequency range. Figure 5(a) shows the relationship between the amplitude of the system (ξ � 0.03, α � 0.3) response at diferent f and the excitation frequency. As the nonlinear coefcient increases, the maximum amplitude of the response decreases, but the degree of curvature does not change. Te upward jump point is also getting smaller. Figure 5(b) shows the relationship between the amplitude response of the nonlinear system (ξ � 0.03, f � 0.5) at diferent z and the excitation frequency. Note that as the nonlinear coefcient increases, the bend of the curve does not change appreciably, but the upward jump point also decreases. Figure 5(c) shows the relationship between the amplitude of the system response (f � 0.5, α � 0.3) with diferent damping ratio ξ and the excitation frequency. With the increase of the damping ratio, the bending degree of the curve does not change, and the upward jump point is also unchanged.

Experiment Setup.
Based on the above discussion, experiments are designed to verify the attenuating efect of NSBD on the vibration response of frame structure and study the infuence of the relevant parameters on the vibration response of frame structure. Te main component of the damper is a buckling beam made of spring steel, which can reach diferent camber heights depending on an external force. Te clamped-clamped buckling beam is fxed within a steel frame supported by four circular steel columns. To better enable the bistable efect, the attached mass is added to the midspan of the buckling beam. Te whole experimental device is fastened on the shaking table using bolts. Under the transverse acceleration provided by the shaking table, the mass block is used to provide the driving force according to F � ma to cause the device to switch between the two equilibrium positions and thus provide vibration reduction. Te experimental device is shown in Figure 6. Te experiments are conducted with and without the NSBD coupled with the frame structure. A data acquisition system is (DH5922D dynamic signal test and analysis system) connected to a computer recorded data from a laser displacement sensor that measures the movements of the frame. Te setup of the experiment and instrumentation is shown in Figure 6. As shown in Figure 6, before installing the NSBD system, holes are reserved in the center of the beam and the square frame, respectively. In the experiment, the Shock and Vibration buckling beam is guided freely through the clamping parts of the square frame at both ends. Ten, prepressure is applied to both ends of the buckling beam to make the buckling beam reach the specifed height, and the corresponding mass blocks are bolted in the middle of the buckling beam according to the experimental requirements after the two ends of the buckling beam are fxed; then the NSBD system is completed.
Te buckling beam in the experiment is arched using spring steel with a prepress at the end, and a mass composed of a square iron block is fxed in the middle of the span. Te mass of the block can be changed to adjust the characteristics of the NSBD. In the experiment, the spring steel is restrained by a square steel tube to control the height of the buckling beam arch. Te support system of the frame structure consisted of four round steel columns that are fxed on the shaking table with bolts. Te experimental parameters are listed in Table 1.

Experimental Conditions.
In the experiment, the efects of the arch-span ratio (i.e., the ratio of arch height to the buckling beam span) and the mass ratio (i.e., the ratio of additional mass block mass to the frame mass) on the vibration reduction performance of NSBD are considered. Te arch-span ratio and mass used in the experiment are listed in Tables 2 and 3.   Shock and Vibration Te experiment validated the attenuating efect of NSBD on the frame structure under sinusoidal and earthquake excitation. Firstly, the shaking table is used to perform a frequency sweep analysis of the experimental frame structure. Te analysis determined that the natural frequency of the experimental frame structure is 2.89 Hz. Based on the switch threshold force of the buckling beams for diferent thicknesses, the amplitude of sinusoidal excitation applied to the frame structure is 0.05 cm. Secondly, the experiments used sinusoidal excitations to examine the damping provided by the NSBD for frame structures of diferent thicknesses. Finally, a total of 10 diferent earthquake excitations selected from two types of sites are used to investigate the vibration control efectiveness of the NSBD at the best damping. According to the code (GB5011-2010) [51], 10 diferent earthquake excitations are selected, as listed in Table 4, which represent diferent site conditions.

Vibration Control Efectiveness under Sinusoidal
Excitation. Sweep sine excitations (0.05 displacement) ranged between 2.78 Hz and 3.0 Hz at 0.1 Hz intervals are used to test the efectiveness of the NSBD at and near the natural frequency. Te frequency-displacement responses of the frame structure for the diferent confgurations of the NSBD are shown in Figure 7 and the corresponding time history responses are shown in Figure 8.
As shown in Figure 7, the inclusion of the NSBD reduced the vibrations of the frame structure especially for those under δ � 0.2 mm, μ � 2.21%, and h � 35 mm. Te installation of the NSBD shifted the natural frequency far away from the dominant excitation frequency, thus providing a great damping efect between 2.86 Hz and 3.0 Hz. Te vibration of frame structure can therefore be suppressed efectively by

Shaking table
Frame structure Data acquisition system NSBD system Figure 6: Experimental device.     Shock and Vibration 9 utilizing the NSBD, which not only mitigates vibrations an overall wide frequency range but also shifted the natural frequency far from the dominant frequency. Figure 8 illustrates the time-history response of frame structure with diferent parameters, and all detailed results are compared in Table 5. Table 5 shows that the maximum attenuation of the peak displacement reached up to 70.77%. Among the diferent tested confgurations, the damping efect is best when the thickness of NSBD is 0.2 mm. An explanation is that when the thickness of the NSBD is large, a larger driving force is needed to initiate the snap-through motion between the two equilibrium positions of the NSBD. Terefore, under the same external excitation, NSBD with a lower thickness is more prone to snap-through motion, which improves vibration attenuation. Tis improvement can be seen in Figure 7(a) vs Figure 7(c) and Figure 7(b) vs Figure 7(d). For the same beam thickness, the diference in arching height leads to signifcant diferences in kinetic energy consumed when snap-through occurs. In these experiments, NSBD with a large arch height has a larger displacement time history, which indicates that the vibration mitigation efect is also more obvious, as shown in Figures 7(a) and 7(b), the same in Figures 7(c) and 7(d). Te attenuating efect of NSBD on the frame structure is more intuitively derived from its displacement time history, as shown in Figure 8, and its experimental regulation is similar to that in Figure 7.

Control Efectiveness under Earthquake Excitation.
Te values of the parameters for the NSBD are selected by the vibration suppression efect of the NSBD under sinusoidal excitation in the shake table tests. It is noted from Figures 7(a) and 7(b) that the vibration suppression efect of the NSBD is better when its mass ratio is 2.21%. As shown in Figures 7(a) and 7(c), the vibration suppression efect of the NSBD is better when its thickness and height are 0.2 mm and 35 mm, respectively. Te frst set of tests uses six earthquake records selected according to Site classifcation I. To investigate the NSBD efectiveness with other site classes, four other earthquake records are selected according to Site classifcation II. Te details of the selected earthquake excitations are listed in Table 4. Figure 9 shows the displacement time histories of the frame structure (with and without optimized NSBD) during diferent earthquake excitations. Te displacements of the  frame structure are generally reduced when the NSBD is installed. Te peak vibration reduction ratio (P) and RMS vibration reduction ratio are both dependent on seismic motions, demonstrating that NSBD can improve the structure's resilience against hazardous earthquake ground motion. Te peak vibration reduction ratio and RMS vibration reduction ratio can be obtained by equations (22) and (23), compared in Table 6. Table 6 shows that the peak damping ratio and average damping ratio can be attenuated respectively by up to 44.63% and 45.03%, demonstrating that the NSBD efectively suppresses the action of earthquake ground motion on the frame structure.
where P refers to the peak vibration reduction ratio (%), a max 1 refers to the maximum displacement amplitude of the structure during the earthquake excitation when the NSBD is installed (mm), a max 2 refers to the maximum displacement amplitude of the structure during the earthquake excitation (mm).

. Conclusions
Te negative stifness bistable damper (NSBD) is currently proposed to dampen the vibration response of the frame structure. Te constitutive equation of the NSBD is derived to analyze the efects of stifness ratio, arch-span ratio, and damping ratio on its restoring capabilities. To verify the vibration suppression performance of the NSBD, shaking table tests of a frame structure are conducted under diferent sinusoidal and seismic excitations with and without an NSBD. Te results reveal that the NSBD can efectively restrain the displacements of the structure. In addition, the buckling beam cannot have signifcant defection to allow it to vibrate between two equilibrium positions, meaning that the beam must remain in its elastic range for the NSBD to be efective. Te main fndings in the present study are summarized as follows: Te proposed damper can greatly mitigate the peak displacement of the frame structure at the dominant frequency and has a wide damping bandwidth. Furthermore, the control efectiveness of the NSBD is robust against a range of sinusoidal and seismic excitations, demonstrating that the NSBD can strengthen structural security against hazardous seismic events.
Tis paper only explored the attenuating efect of NSBD on a single-story frame structure experimentally. Future works will be undertaken to perform related analytical and numerical research. Furthermore, the vibration control of structures using the NSBD in actual engineering will be further studied and improved in future work.

Data Availability
Te data used to support the fndings of this study are included in the article.

Conflicts of Interest
Te authors declare that they have no conficts of interest.